Math Problem Statement
Solution
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The image appears to involve identifying whether each system of linear equations, represented in reduced row echelon form (RREF), has a unique solution, infinitely many solutions, or no solutions. I'll solve these problems step by step.
Step 1: Analyze the first matrix
The matrix provided in the first problem is:
1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ This matrix corresponds to the following system of equations: 1. $$x_1 = 0$$ 2. $$x_2 + 2 = 0 \implies x_2 = -2$$ 3. $$x_3 + 3 = 0 \implies x_3 = -3$$ - The fourth row $$0 = 0$$ is consistent, so it does not add constraints. - The system has exactly one solution: $$(x_1, x_2, x_3) = (0, -2, -3)$$. **Answer for Problem 1:** **Unique solution**. --- ### Step 2: Analyze the second matrix The matrix for the second problem is: $$\begin{bmatrix} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ This corresponds to: 1. $$x_1 = -2$$ 2. $$x_2 = 4$$ - $$x_3$$ is a free variable (there is no pivot in the third column). - This means the system has infinitely many solutions, which depend on $$x_3$$: $$(x_1, x_2, x_3) = (-2, 4, t), \text{ where } t \in \mathbb{R}.$$ **Answer for Problem 2:** **Infinitely many solutions**. --- ### Step 3: Analyze the third matrix The third matrix is: $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 2 \end{bmatrix}$$ This corresponds to: 1. $$x_1 = 0$$ 2. $$x_2 + 2 = 0 \implies x_2 = -2$$ 3. $$x_3 + 4 = 0 \implies x_3 = -4$$ 4. The fourth row $$0 = 2$$ is inconsistent. - Because of the inconsistent row, the system has **no solutions**. **Answer for Problem 3:** **No solutions**. --- ### Step 4: Analyze the fourth matrix The fourth matrix is: $$\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ This is the trivial case where no constraints exist. Any values of $$x_1, x_2, x_3$$ satisfy the system. **Answer for Problem 4:** **Infinitely many solutions**. --- ### Final Summary of Answers: 1. **Unique solution** 2. **Infinitely many solutions** 3. **No solutions** 4. **Infinitely many solutions** Would you like detailed steps for any specific problem? Here are some related questions: 1. What is the significance of a free variable in a system of equations? 2. How do you determine if a matrix is in reduced row echelon form? 3. What does an inconsistent row in a matrix indicate about the system of equations? 4. How can you use row reduction to solve a system of equations? 5. Can a matrix in RREF have both infinitely many and no solutions? **Tip:** Always check the consistency of a system by observing the rows of zeros in the augmented matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Reduced Row Echelon Form (RREF)
Solution Types
Formulas
Gaussian elimination
Matrix consistency rules
Theorems
Rank-Nullity Theorem
Existence and Uniqueness Theorem for Linear Systems
Suitable Grade Level
Undergraduate Level (Introductory Linear Algebra)
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